was possible to extract the binding energy to
these kinetically stabilized non-native structures.
By using a DNA hairpin with two binding sites
separated by 4 bp, we observed the formation
of a misfolded structure consisting of two short
(4 bp) hairpins in series (Fig. 3D, hairpin M). Such
an off-pathway kinetic state is unobservable
in the absence of ligand owing to its low energy
of formation; however, it is kinetically stabilized by the binding of the ligand. We applied
Eq. 2 by choosing a starting point of the cyclic
protocol where the native-bound and misfolded-bound conformations are distinguishable (F

10 pN) and found that the energy of binding
to the native configuration is the same as the
energy of binding to the misfolded configuration ðDGbind;M−N ¼ 2 T 1kB T Þ (fig. S10 and tables
S15 and S16).

In this work, we have introduced a fluctuation
theorem for ligand binding to directly determine binding energies as a function of ligand
concentration in single-molecule experiments.

Using different biomolecular systems of increas-ing complexity, we provided a single-moleculeverification of the law of mass action and showedhow the FTLB can account for mass exchangebetween a molecular system and the environ-ment. We could resolve binding energies to spe-cific and nonspecific sites with affinities spanningsix orders of magnitude. The FTLB provided adirect experimental measurement of bindingenergies without assuming any model or reac-tion scheme, which is particularly useful in caseswhere the law of mass action does not hold. Toshow this, we applied the FTLB in two situationswhere this may happen: the cooperative bindingof multiple ligands to the same substrate andthe stabilization of kinetic structures throughligand binding, both of which are measurements

AB

0.1 1 10
[Echinomycin] (µ M)

-5
0

5
10

ΔG
b
i
n
d
(
kB T
)

0
0.2
0.4
0.6
0.8

6 9 12 15 18 21 24
ρ(
F
)

First rupture force (pN)
SP NSP

C

T
A
T
A
T

G
C

A
T
C
C
C

5’

A
T
A
T
A

C
G

T
A
G
G
G

3’
G A
A A

TATATCGATCCC5’ATATAGCTAGGG3’G AA Abinding site

A
T
A
T
G
C
A
T
G
C
A
T
C
C
C

5’

T
A
T
A
C
G
T
A
C
G
T
A
G
G
G

3’
G A
A A

A
T
G
C
A
T
A
T
G
C
A
T
C
C
C

5’

T
A
C
G
T
A
T
A
C
G
T
A
G
G
G

3’
G A
A A

no site

D

ΔG0SP=20.0±0.8 kBT

ΔG0NSP= 13.2±0.5 kBT

5
10
15
20
25

F
o
r
ce
(p
N
)

No ligand

300 nM Echinomycin

C NC

A
T
A
T
G
C
A
T
A
T
A
T
G
C
A
T
C
C
C
5’
T
A
T
A
C
G
T
A
T
A
T
A
C
G
T
A
G
G
G
3’
G A
A A
M

0
0.2
0.4
0.6

−9 −6 −3 0 3 6 9ρΔGbind per ligand (kBT)

5’CCCTTGCAACGTATAGAAATATTGCAACGTAGGG-3’A TT AA T

NC

(1 bound)

NC(2 bound)

C(2 bound)

50 nm

λ

Fig. 2. Oligonucleotide binding to DNA. (A) Scheme
of N, U, and B states. (B) Cyclic pulling curves that
start and end at low forces (~6 pN) classified according to their initial (blue dot) and final (cyan dot)
state. (C) Partial work distributions of B → N (green)
and N → B (magenta) transitions. (D) Binding energy of the 10-base oligonucleotide (blue) and fit to
the law of mass action (red line). The value obtained
from hopping equilibrium experiments at [oligo-nucleotide] = 400 nM is shown in cyan (see section
S1.7 of the supplementary materials and methods).
(D, inset) Contribution of the ratio fN→B=fB→N to the
binding energy. Error bars were obtained from bootstrap as described in Fig. 1.

Fig. 3. Binding specificity, allostery, and kinetic
stabilization of misfolded states for the peptide
echinomycin. (A) First rupture force distribution
of hairpins SP (red) and NSP (blue) in the absence
(light color) and presence (dark color) of echinomycin. (B) Binding energy of echinomycin to a specific (red) and nonspecific (blue) site and fit to the
law of mass action. (C) Hairpins NC and C contain
two specific binding sites (red boxes) separated by
2 bp or placed contiguously, respectively. Binding
energy per ligand when one (blue) or two (green)
ligands are bound to hairpin NC and when two
ligands are bound to hairpin C (magenta) (
[echinomycin] = 3 mM). Gaussian distributions are reconstructed from mean and variance of measurements.
The wider distribution for the single bound state
in hairpin NC (blue) is due to the lower number of
paths reaching this state at high ligand concentration, increasing measurement error. (D) Pulling
cycle of hairpin M in the presence of echinomycin.
The unfolding curve (blue) shows two force rips at
F 20 pN corresponding to the unbinding of two
ligands bound to specific sites. In the refolding curve
(cyan), the hairpin does not fold back to the native
state and misfolds into a kinetically stabilized configuration of longer molecular extension (~40% of
refolding curves at [echinomycin] = 10 mM). Error
bars were obtained from bootstrap as described
in Fig. 1.